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Delayed population models with Allee effects and exploitation
1. | Departamento de Matemática Aplicada II, E.T.S.E. Telecomunicación, Universidade de Vigo, Campus Marcosende, 36310 Vigo |
2. | Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged, Hungary |
References:
[1] |
D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theoret. Biol., 218 (2002), 375-394.
doi: 10.1006/jtbi.2002.3084. |
[2] |
B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model, Math. Biosci., 248 (2014), 78-87.
doi: 10.1016/j.mbs.2013.12.003. |
[3] |
C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources, $2^{nd}$ edition, John Wiley & Sons, Hoboken, New Jersey, 1990. |
[4] |
F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008.
doi: 10.1093/acprof:oso/9780198570301.001.0001. |
[5] |
A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907-12912. |
[6] |
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[7] |
S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Biol. Dyn., 4 (2010), 397-408.
doi: 10.1080/17513750903377434. |
[8] |
S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?, J. Math. Biol., 65 (2012), 1411-1415.
doi: 10.1007/s00285-011-0496-3. |
[9] |
K. P. Hadeler, Neutral delay equations from and for population dynamics, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1-18. |
[10] |
C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114.
doi: 10.1016/j.jde.2013.12.015. |
[11] |
A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54. |
[12] |
A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164-224. |
[13] |
M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor. Ecol., 7 (2014), 335-349.
doi: 10.1007/s12080-014-0222-z. |
[14] |
T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95.
doi: 10.1007/s10998-008-5083-x. |
[15] |
T. Krisztin and E. Liz, Bubbles for a class of delay differential equations, Qual. Theory Dyn. Syst., 10 (2011), 169-196.
doi: 10.1007/s12346-011-0055-8. |
[16] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. |
[17] |
E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting, Theor. Ecol., 3 (2010), 209-221.
doi: 10.1007/s12080-009-0064-2. |
[18] |
E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70. |
[19] |
E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224.
doi: 10.3934/dcds.2009.24.1215. |
[20] |
E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback, J. Differential Equations, 255 (2013), 4244-4266.
doi: 10.1016/j.jde.2013.08.007. |
[21] |
E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622.
doi: 10.1137/S0036141001399222. |
[22] |
J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128.
doi: 10.1007/BF01790539. |
[23] |
G. Röst, On the global attractivity controversy for a delay model of hematopoiesis, Appl. Math. Comput., 190 (2007), 846-850.
doi: 10.1016/j.amc.2007.01.103. |
[24] |
G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.
doi: 10.1098/rspa.2007.1890. |
[25] |
S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications, NATO Sci. Ser. II Math. Phys. Chem., Springer, Dordrecht, 205 (2006), 477-517.
doi: 10.1007/1-4020-3647-7_11. |
[26] |
S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.
doi: 10.1007/s002850000070. |
[27] |
S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[28] |
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers, Dordrecht, 1997.
doi: 10.1007/978-94-015-8897-3. |
[29] |
A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications, Kluwer Academic Publishers, Dordrecht, 1993.
doi: 10.1007/978-94-011-1763-0. |
[30] |
D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.
doi: 10.1137/0135020. |
[31] |
A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries, Math. Biosci., 232 (2011), 66-77.
doi: 10.1016/j.mbs.2011.04.004. |
[32] |
T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.
doi: 10.1098/rspa.2009.0650. |
show all references
References:
[1] |
D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theoret. Biol., 218 (2002), 375-394.
doi: 10.1006/jtbi.2002.3084. |
[2] |
B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model, Math. Biosci., 248 (2014), 78-87.
doi: 10.1016/j.mbs.2013.12.003. |
[3] |
C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources, $2^{nd}$ edition, John Wiley & Sons, Hoboken, New Jersey, 1990. |
[4] |
F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008.
doi: 10.1093/acprof:oso/9780198570301.001.0001. |
[5] |
A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907-12912. |
[6] |
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[7] |
S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Biol. Dyn., 4 (2010), 397-408.
doi: 10.1080/17513750903377434. |
[8] |
S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?, J. Math. Biol., 65 (2012), 1411-1415.
doi: 10.1007/s00285-011-0496-3. |
[9] |
K. P. Hadeler, Neutral delay equations from and for population dynamics, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1-18. |
[10] |
C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114.
doi: 10.1016/j.jde.2013.12.015. |
[11] |
A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54. |
[12] |
A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164-224. |
[13] |
M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor. Ecol., 7 (2014), 335-349.
doi: 10.1007/s12080-014-0222-z. |
[14] |
T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95.
doi: 10.1007/s10998-008-5083-x. |
[15] |
T. Krisztin and E. Liz, Bubbles for a class of delay differential equations, Qual. Theory Dyn. Syst., 10 (2011), 169-196.
doi: 10.1007/s12346-011-0055-8. |
[16] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. |
[17] |
E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting, Theor. Ecol., 3 (2010), 209-221.
doi: 10.1007/s12080-009-0064-2. |
[18] |
E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70. |
[19] |
E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224.
doi: 10.3934/dcds.2009.24.1215. |
[20] |
E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback, J. Differential Equations, 255 (2013), 4244-4266.
doi: 10.1016/j.jde.2013.08.007. |
[21] |
E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622.
doi: 10.1137/S0036141001399222. |
[22] |
J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128.
doi: 10.1007/BF01790539. |
[23] |
G. Röst, On the global attractivity controversy for a delay model of hematopoiesis, Appl. Math. Comput., 190 (2007), 846-850.
doi: 10.1016/j.amc.2007.01.103. |
[24] |
G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.
doi: 10.1098/rspa.2007.1890. |
[25] |
S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications, NATO Sci. Ser. II Math. Phys. Chem., Springer, Dordrecht, 205 (2006), 477-517.
doi: 10.1007/1-4020-3647-7_11. |
[26] |
S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.
doi: 10.1007/s002850000070. |
[27] |
S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[28] |
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers, Dordrecht, 1997.
doi: 10.1007/978-94-015-8897-3. |
[29] |
A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications, Kluwer Academic Publishers, Dordrecht, 1993.
doi: 10.1007/978-94-011-1763-0. |
[30] |
D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.
doi: 10.1137/0135020. |
[31] |
A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries, Math. Biosci., 232 (2011), 66-77.
doi: 10.1016/j.mbs.2011.04.004. |
[32] |
T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.
doi: 10.1098/rspa.2009.0650. |
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